Is the universe $p$-adic?

Question

(I'm a mathematician in my training, not a physicist)

In all the lectures and popular science shows that I have seen, the model of the expansion of the universe is such that no matter where you are, you see everyone moving away from you, as if you're in the center of the universe.

This seems an awfully similar to the metric of $p$-adic fields, rather than that of the real numbers. In other words, this is similar to what ultrametrics model, where every point is the center of every ball in which it lies.

Since I guess it does make sense for the universe to be a vector space over some field, are there any serious models of the universe as a $p$-adic space, rather than a Euclidean or Riemannian space?

Comments

You may be interested in this paper of Susskind et al: http://arxiv.org/abs/1110.0496

---

The universe may or may not be p-adic at some level, but this is not a good argument for it... Someone on the surface of Earth also sees themselves at the center of things, everything visible contained within a horizon that is just as far away in every direction. But that doesn't mean that the surface of the Earth is p-adic.

Answer 1

Spatial topologies are completely different from p-adic topologies in that they have the property of intersecting balls: you can have a ball at x and a ball at y with an intersection which is not equal to either ball. In p-adic balls, if two balls intersect, one is entirely inside the other. P-adic spaces are like infinite trees going down, while on the other hand, physical space is an overlapping structure, where balls overlap.

In the loop literature, where people attempt to build metrics from graphs, sometimes you do end up producing a p-adic like metric, where the intersection of ball A and ball B is either equal to ball A or ball B. This property makes it that the distance function is unphysical, it doesn't correspond to the overlap property of balls in space.

The only case in which you can get a long-distance disconnection which can be modelled as tree-like separation is if you consider the extremely large-scale structure of an eternally inflating universe, so that each little point is modelling an independent separate causal patch. This seems to be the picture you are getting at, but this picture is difficult to turn into physics, because we can only make observations within one patch, so you need to formulate the predictions in one patch for the idea to make sense in positivism.

Comments

Well, if quantum mechanics seem to show that objects on a very very small scale behave somewhat differently than objects on a "normal" and "large" scale; why wouldn't it be reasonable that objects on a huge, tremendous, scale behave differently as well? Objects like portions of the universe? Perhaps a multiverse, where two "bubbleverse" intersect, then one is a sub-bubble of the other.

Something like that... Mathematically speaking, that sounds reasonable to me.

---

Yes, this is the best interpretation I could give the statement, that you are talking about how disconnected universes nest inside one another. The problem is that this is a multiverse property, it isn't true for past light-cones inside a causal patch, past light cones can intersect without being contained one in the other. Since it is a multiverse property, you need to be able to link it to some observation, and I don't see how to do this in any way. It's not like you gave a method for predicting statistics of our universe from the hypothetical p-adic like structure of the multiverse.

Answer 2

Some important papers on p-adic quantum mechanics (though nothing of real value came out of it):

• http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104178988
• http://link.springer.com/article/10.1134%2FS1063778810020055 (a free copy is available from 144.206.159.178/FT/8304/650261/13044559.pdf)
• http://arxiv.org/abs/0904.4205

Comments

Suggest edit: link the arxiv paper to its abstract page http://arxiv.org/abs/0904.4205

---

I prefer to see links to the pdf's as it saves some steps to get the real information. But I don't mind if someone else enforces a different style.

---

The problem is ArXiV PDFs are temporary files and may disappear or be moved. Also, linking to a PDF instead of the abstract page, means you are "freezing" the version - The authors may subsequently reissue an improved version, or withdraw the paper if it turns out to be wrong. Nor will the reader be able to see and quickly access anciliary files if any, or read the PostScript version if there is one. So in summary, linking to ArXiv PDF files directly is very very naughty and undesirable!

---

OK, I changed the link.

---

Hm, it seems you've accidentally changed only the displayed link, but the url remains the same. I've fixed it.  

---

@JiaYiyang Just a note: You can suggest edits by clicking the "suggest edits" button below the answer. It's ok now of course. I'll let Arnold review your edit himself in this case.  

---

@dimension10: It led me to a weird page. I'm confused about what to do on that page.